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For a particular generalised-NFW (gNFW) mass distribution, we derive analytical expressions for its surface mass density, projected mass, and deflection potential, given in terms of three representations of the Fox H-function, for which we provide their corresponding power (logarithmic) series expansions. They are handy in computationally intensive tasks. From these results, we obtain closed-form expressions for the super-NFW (sNFW) in terms of complete elliptic functions, which have not yet been reported in the literature. Additionally, we find that, for a fixed ₀ (characteristic convergence), when the number of images is maximal (three for the sNFW), the sum of their signed magnification, named Iₒ₍₅ₖ, varies with the position of the source y inside the radial caustic y₂₂. The higher variations occur for ₀ 1, while Iₒ₍₅ₖ can be considered as constant when ₀ is approximately within the range 2-10. This range can be extended depending on the observational uncertainties, since for higher values of ₀, the variations in Iₒ₍₅ₖ are relatively small. This behaviour is shared with the NFW and Hernquist lenses.
Torres-Ballesteros et al. (Tue,) studied this question.